This question is concerned with the algebraic side of the theory of prolongations as explained in this paper by V. Guillemin and S. Sternberg. Let me first introduce my notation.
We're working with a ...

Say you have a graded module $M$ over a field $k$. In my mind I have $M=k[x,y,z]/(y^2-xz)$ graded by rank. The Hilbert series of this is $\sum_{n=0}^{\infty} (2n+1)x^n$. You can represent this as a ...

It must be a very basic question, but I just can't figure out...
Let $P$ be a graded $A$-module ($A$ is a commutative associative with unity). Can $P$ have two different direct decompositions, that ...

Let $(A,d)$ be a differential graded commutative and associative algebra.
A derivation on $A$ is a linear endomorphism $L: A \to A$, that satsfies
$L(ab)= L(a)b+ aL(b)$. More general a derivation of ...

Let $A^{\bullet}$ be a graded commutative algebra. Denote by $A^{\bullet}$-mod the category of graded modules over $A^{\bullet}$. Let $A$ be $A^{\bullet}$ considered as an algebra (we forgot grading). ...

Algebraic geometry offers some properties and criteria for homomorphism of commutative rings to be flat. What about homomorphisms of graded-commutative rings? You can define flatness as usual: $R \to ...

I am looking for reference books or papers on graded modules over the polynomial ring $k[x_0, \ldots, x_n]$. Any good commutative algebra text like Eisenbud's Commutative Algebra already contains a ...

I'm reading Commutative Algebra by Bourbaki, and I'm on chapter III of the book. I really think that the author made a typo there, but I'm not so sure. It's Proposition 1, and its Corollary on page ...

Assume $R$ is a graded ring and the $M_i$ are graded modules. Then Bruns and Herzog define the graded direct product $^*\Pi M_i$ as the submodule of $\Pi M_i$ generated by the sequences $(x_i)$ with ...

I am trying to calculate the Poincaré series $P(M,t)$ with respect to the standard degree grading of the graded $\mathbb{C} [x,y,z,w]$-module $ M=\mathbb{C}[x,y,z,w]/I$, where $I = (x,w) \cap (z,w) ...

Wikipedia says that a graded $A$-algebra is just a graded $A$-module that is also a graded ring.
Question: when one says then "finitely generated graded $A$-algebra", does one mean that every element ...

How can we show that $\dim R/p=0\Leftrightarrow p=(x_{1},\ldots,x_{n})\Leftrightarrow R/p\simeq\mathbb{K}$, where $R=\mathbb{K}[x_{1},\ldots,x_{n}]$ is considered graded with standard grading (i.e. ...

$\def\finiteprod#1#2{#1_{1}\times#1_{2}\times\dots\times#1_{#2}}$In Lang's algebra he defines the tensor product as a universal object
in the category of multilinear maps from $\finiteprod En$ where ...